Question

Find the equation of the perpendicular bisector of the line segment joining the points (2, 3) and (6, −5).

Answer 1

⇒ The perpendicular bisector passes through the midpoint of the segment joining  (2, 3) and (6, −5), using the mid-point formula:

`M = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

`M = ((2 + 6)/2, (3 + (-5))/2)`

`M = (8/2, (-2)/2)`

∴ M = (4, −1)

⇒ Using the slope(m₁) formula with points (2, −5) and (0, −3):

`m = (y_2 - y_1)/(x_2 - x_1)`

`m_1 = (-5 - 3)/(6 - 2)`

`m_1 = (-8)/4`

∴ m₁ = −2

⇒ Since the bisector is perpendicular, its slope (m₁) is the negative reciprocal of m₁:

`m_2 = -1/m_1`

`m_2 = - 1/-2`

∴ `m_2 = 1/2`

⇒ Using the point-slope formula with `m = 1/2` with point (4, −1):

y − y₁ = m(x − x₁)

`y - (-1) = 1/2 (x - 4)`

`y + 1 = 1/2 (x - 4)`

2(y + 1) = x − 4

2y + 2 = x − 4

⇒ Rearranging the above equation in the standard form (Ax + By + C = 0),

x − 2y − 4 − 2 = 0

x − 2y − 6 = 0

Hence, the equation of the perpendicular bisector is x − 2y − 6 = 0.